Distributivity in Incompletely Specified Type Hierarchies : Theory and Computational Complexity †

Type hierarchies are a central feature of current systems which employ feature structures for the representation and manipulation of linguistic knowledge, such as TFS, ALE, CUF, and LKB. The system CUF stands out amongst these in its very general language for describing such hierarchies. While the other systems use a description language which allows only the specification of meets (greatest lower bounds of types), CUF provides a very general type definition language which allows a full complement of Boolean operations. This flexibility permits much simpler and more natural specifications of certain types of knowledge representation which occur naturally in paradigms such as HPSG. However, along with this increased power of expressiveness comes an increase in computational complexity. Specifically, while the simpler type systems are in fact so restrictive that any admitted specification is semantically meaningful, the power of the CUF type-specification language results in highly nontrivial decision question about well-formedness. These well-formedness questions revolve around the property of distributivity, as any semantically meaningful hierarchy — that is, one in which meet represents intersection of object classes and join represents union — must be distributive. The contribution of this report is twofold. First of all, we provide a very general development of extension problems for partial type hierarchies to full distributive hierarchies, which is exactly the problem one needs to solve to determine wellformedness and other properties of type hierarchies in the CUF spirit. Secondly, we analyze the computational complexity of deciding the existence of various forms of distributive extensions. We show that under relatively weak assumptions concerning the height and fanout of the partial specification, all of the various decision problems are NP-complete. However, in the process of developing the formalism necessary for this detailed analysis, we also identify some close connections to certain kinds of satisfiability problems in propositional logic, which will hopefully prove useful in identifying useful special cases in which the computational problems are tractable.